Theory Of Equations Research Articles

A mechanism for detecting normally hyperbolic invariant tori in differential equations

Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the first non-vanishing higher order averaged function. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.

Quantum spin helices more stable than the ground state: Onset of helical protection

Topological magnetic structures are promising candidates for resilient information storage. An elementary example is spin helices in one-dimensional easy-plane quantum magnets. To quantify their stability, we numerically implement the stochastic Schr\"odinger equation and time-dependent perturbation theory for spin chains with fluctuating local magnetic fields. We find two classes of quantum spin helices that can reach and even exceed ground-state stability: spin-current-maximizing helices and, for fine-tuned boundary conditions, the recently discovered ``phantom helices.'' Beyond that, we show that the helicity itself (left or right rotating) is even more stable. We explain these findings by separated helical sectors and connect them to topological sectors in continuous spin systems. The resulting helical protection mechanism is a promising phenomenon for stabilizing helical quantum structures, e.g., in ultracold atoms and solid-state systems. We also identify a third type of phantom helix in the system.

Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the C1,γ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is C1,Dini in space. The free boundary then must immediately become C1,γ for a universal γ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.

Global dynamics of a stochastic reaction–diffusion predator–prey system with space-time white noise

This paper proposes a stochastic reaction–diffusion predator–prey system with fear under the interference of space-time white noise by using the related theories of stochastic partial differential equations. The motivation of this paper is twofold: (i) mathematically, to try to investigate the effects of environmental noise and diffusion on population dynamics; (ii) biologically, to study how environmental noise affects the permanence of species. First, we analyse the well-posedness of solutions. Then sufficient conditions for extinction and permanence of prey and predator populations are derived. Moreover, we present the existence and uniqueness of stationary distribution of this system. Finally, based on numerical simulations and theoretical analysis, it is revealed that (i) high-intensity white noise can lead to the extinction of predator populations; (ii) low-intensity white noise may prolong the period of periodic solutions. This study provides new theoretical guidance for exploring ecological issues in the face of environmental disturbance and spatial diffusion.

Study on the hydrodynamic added mass of rectangular pier under sine wave actions

As for deep-water bridges in river, sea, and reservoirs, the water-structure interaction is non-negligible when investigating its seismic performance. Morison equation and radiation wave theory are two main methods for calculation of hydrodynamic pressure resulted from water-structure interaction, which cannot truly simulate the complicated interaction between water and structures. This study conducts a series of underwater shaking table tests on seven rectangular steel tubes to predict the hydrodynamic added mass of rectangular piers considering water-pier interaction. The influence of action amplitude, action frequency, water depth, depth of cross section, and width of upstream face of the pier on hydrodynamic added mass is investigated. The results show that the hydrodynamic added mass of pier first decreases and then increases with action frequencies. The hydrodynamic added mass is proportional to a-th (a > 1) power of water depth due to the influence of structures on the movement of water. The hydrodynamic added mass increases with depth of cross section and width of upstream face of pier, and changes slightly with the action amplitude with maximum percentage of 10% and thus can be neglected. Therefore, except for the water depth, depth of cross section, and width of upstream face, the frequency characteristics of ground motions should be considered when analyzing the water-pier interaction under earthquake actions.

Determining the fundamental dynamics of global sustainability in the Anthropocene from a vulnerability perspective

This paper conducts a detailed quantitative study of the indicated long-term levels, nature and dynamics of sustainability at the global spatial scale over time, within the context of the vulnerability of the environment-human relationship and system. In this study, the Sustainability Dynamics Framework is applied to the results of the Notre Dame Global Adaptation Initiative index for the period 1995–2018. The study indicates that the obtained S-values at the global spatial scale throughout the specified period were consistent with very weak unsustainability occurring. Through statistical analysis, the study indicates that the triumvirate of influencing factors (Population, GDP, Greenhouse Gases) have a significant influence and impact individually and cumulatively upon obtained S-values. The study then determines the net change in obtained E, HNI and S-values, and consequently evaluates the dynamics of the rate of change of the Earth System ( dE/dt), as defined in the Anthropocene Equation theory. To support this, Fourier analysis of net change of S-values is applied for the first time within the SDF application methodology, using eight time periods of 16 years’ duration over the period 1996–2018. The analysis indicates continuous overlapping cycles of 7 years’ duration, consisting of a lag time of 3 years leading to an impact period of 4 years. The Fourier analysis potentially indicates clearly for the first-time humanity’s footprint signal upon the environment-human relationship and system in the Anthropocene. The study concludes that humanity’s actions and inactions have resulted in the indicated unsustainability and vulnerability of the environment-human relationship and system. As a consequence, there is a potential realistic risk of systemic collapse unless fundamental changes occur in respect to the global environment-human relationship.

Research on the Reliability of a Two-Robot Security System with Early Warning Function

In this paper, the mathematical model of a kind of two-robot security system with an early warning function is studied. By using strongly continuous operator semigroup theory and Volterra integral equation theory, the properties of the semigroup of the system operator, the existence and uniqueness of nonnegative solution, and the well-posedness of solution are discussed, respectively. Under the assumption that the failure rate and repair rate of the system are constants, the equations of the robot system are transformed into an ordinary differential equation group, and then the instantaneous reliability and stable-state reliability of the system are obtained. The reliability and zero-state controllability of the system are proved. Finally, the numerical solution of the system model is obtained by using MATLAB mathematical software, and the corresponding numerical simulation diagram is given. The results show that the conclusions of numerical calculation and numerical simulation are in accordance with the results of reliability theory, thereby the reliability of the robot safety system is verified.

Functional Education as a Nexus between Agricultural and Industrial Revolution: An Epidemiological Modeling Approach

Education has been regarded as a major route to economic prosperity. Its potentials, if properly exploited, have the tendency to revolutionize a nation’s productivity, income, and development. [Objetive] In this work, a mathematical model was formulated, and an epidemiological modeling approach was explored to derive the necessary and sufficient conditions for agricultural and industrial transformation in Nigeria in terms of education. The use of mathematics to quantify such a phenomenon is relatively new. [Methodology] Ample mathematics theorems were employed to test for the existence, boundedness, and positivity of the model’s solutions which are basic features of a valid epidemiological model. The model was solved to derive the equilibrium points, and the analytic threshold that governed revolution in industry and agriculture in terms of education was derived. Stability and sensitivity analyses were conducted via the stability theory of nonlinear differential equations and normalized forward sensitivity index, respectively. Simulations were later conducted using the Maple software to validate the analytical results. [Results] A government expenditure of 25% on education with or without corruption did not make education functional. Also, an increase in expenditure on education to 65% but with corruption of more than 30% retarded functional education. In general, education became functional when government investment in education attained 60%, with corruption levels lower than 20%. [Conclusions] Both the analytical and numerical results linked agricultural transformation and industrial revolution through functional education to massive investment in education and victory over corruption.

Asymptotic Study of Divorce Model with Pre-Marriage Preparedness as Control

Peaceful cohabitation in a marriage institution is challenged with separation/divorce because of distinct individual psychological build-up. A deterministic model for the divorce epidemic was proposed using standard incidence as a forcing function. The stability theory of differential equations was used to perform the model analysis qualitatively on which the equilibria obtained are locally and globally stable. Bifurcation and sensitivity analysis of the model were performed; parameters responsible for managing and eradicating the spread of divorce in marriages were determined. A numerical simulation was performed with results that showed pre-marriage preparedness and conscientious growth in tolerance of individual differences as a stabilizer to marriages.

P-Regularity Theory and the Existence of a Solution to a Boundary Value Problem Continuously Dependent on Boundary Conditions

For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are p-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of p-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.

The role of the pressure in the regularity theory for the Navier-Stokes equations

We first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak solutions. Then, an $\varepsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\varepsilon$-regularity criterion with a new approach using a local version of Leray projection operator. As an application of the approach, we obtain the short-time regularity result on a bounded domain for dissipative solutions.

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

TRANSIENT PROCESSES IN ELECTRICAL COMPLEX OF PRODUCTION WELL WITH DOWNHOLE COMPENSATOR OF REACTIVE POWER

Link for citation: Glazyrin A.S., Isaev Yu.N., Kopyrin V.A., Timoshkin V.V., Kladiev S.N., Khamitov R.N., Kovalev V.Z., Losev A.F., Leonov A.P., Popov E.I. Transient processes in electrical complex of production well with downhole compensator of reactive power. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 5, рр.168-180. In Rus. The relevance. At the present time at routine complication of technological processes to increase efficiency of production processes due to introduction of new equipment conflicts between functional capabilities of various technical means and their ability to function with specified quality, in specified electromagnetic environment and do not create inadmissible electromagnetic interference to other technical means. The electrical engineering complex for oil production includes a combination of ground and submersible electrical equipment assembled for receiving, transforming electrical energy, controlling its flow, converting electrical energy into mechanical energy and transmitting it to a submersible pump, including various electrical and electronic devices: a control station with a frequency converter, a step-up transformer, a cable line, a submersible electric motor, a downhole reactive power compensator and other devices. Any change in the layout of electrical systems for extracting oil or technical characteristics of the equipment used requires careful analysis of transient processes between the operating modes of electrical equipment in order to prevent the risks of false triggering of electrical equipment protection devices. The purpose: analysis of transient processes in the electrical system of the production well with the downhole reactive power compensator when the power supply is disconnected in order to prevent the risks of false operation of the compensator protection devices. The object: section of a power supply system for a cluster of oil producing wells equipped with electric centrifugal pump installations with downhole reactive power compensators. Methods: positions of theoretical bases of electrical engineering, theory of electric drive, theory of differential equations, numerical methods. Results. The authors developed the mathematical model of the electrical system of the submersible unit for oil production with a downhole reactive power compensator, which enables to investigate transient processes in start-up and shutdown mode of the submersible induction motor. Characteristics of transient processes are obtained in elements of electrical complex of submersible plant for oil production when power supply is switched on and disconnected. It is established that the maximum current surge in the downhole reactive power compensator is 2,56 times with the parameters of the electrical system of the production well accepted in the study. The inventive measures for reducing the number of false actuations of the protection devices are also disclosed.

General path integrals and stable SDEs

The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \operatorname{d} Z_t=\sigma(Z_{t-})\operatorname{d} X_t driven by a two-sided \alpha -stable Lévy process, in the spirit of the classical Engelbert–Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence and uniqueness of weak solutions for \alpha\in(0,1) . Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We prove integral tests for finiteness of path integrals under minimal assumptions.

Combustion synthesized novel SrAlBO4:Eu3+ phosphor: Structural, luminescence, and Judd-Ofelt analysis

A series of new phosphor SrAlBO4 doped by Eu³+ was synthesized by using very known Combustion method. The structural and optical properties explored using X-ray diffraction (XRD). The XRD pattern was matched well with the standard PDF card no. 1542236.The obtained phosphor had orthorhombic structure with space group Pccn (56). The image of Scanning electron microscope shows the external morphology of SrAlBO4 phosphor material, which revails the irregular morphology and the material showed the non uniform structure with agglomerates' size was size ranging in several micrometers. The confirmation of present element and their percentage also shown in the EDX image. The luminescence properties of rare earth activated SrAlBO4 phosphor were determined by measurement of excitation and emission spectra. The PL emission spectra of Eu3+ doped SrAlBO4 phosphor show characteristics bands at 590 nm & 614 nm which corresponds due to 5D0 →7F1 and 5D0 →7F2 transition of Eu3+ ions upon 395 nm excitation. The effect of different concentration of Eu3+ on luminescence emission intensity of SrAlBO4 phosphor was also studied. Further, concentration quenching is explained using Blasse's equation and Dexter's theory. Also, Judd-Ofelt analysis was performed on photoluminescence emission spectra. On investigation synthesized rare earth activated SrAlBO4 phosphor can be suitable for all lighting application devices.

Propagation of Moments for Large Data and Semiclassical Limit to the Relativistic Vlasov Equation

We investigate the semiclassical limit from the semirelativistic Hartree–Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form , , , and , with the convention if . For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398–433], where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.

On the fundamental qualitative properties of integro-delay differential equations

This paper discusses qualitative properties of solutions of certain unperturbed and perturbed systems of nonlinear integro-delay differential equations (IDDEs), namely asymptotic stability, uniform stability, integrability and boundedness. Here, four new theorems are proved on these properties of solutions by using Lyapunov–Krasovskiǐ functional (LKF) technique. As illustrations and applications of our results, we also provide two examples, solve them numerically, and plot the trajectories of their solutions. The results of this paper include weaker sufficient conditions than the ones found in the literature, e.g., some superfluous conditions are removed here, and the results have also new contributions to the qualitative theory of integro-differential equations (IDEs) and IDDEs.

Isospin 0 and 2 two-pion scattering at physical pion mass using all-to-all propagators with periodic boundary conditions in lattice QCD

A study of two-pion scattering for the isospin channels, $I=0$ and $I=2$, using lattice QCD is presented. M\"obius domain wall fermions on top of the Iwasaki-DSDR gauge action for gluons with periodic boundary conditions are used for the lattice computations which are carried out on two ensembles of gauge field configurations generated by the RBC and UKQCD collaborations with physical masses, inverse lattice spacings of 1.023 and 1.378 GeV, and spatial extents of $L=4.63$ and 4.58 fm, respectively. The all-to-all propagator method is employed to compute a matrix of correlation functions of two-pion operators. The generalized eigenvalue problem (GEVP) is solved for a matrix of correlation functions to extract phase shifts with multiple states, two pions with a non-zero relative momentum as well as two pions at rest. Our results for phase shifts for both $I=0$ and $I=2$ channels are consistent with and the Roy Equation and chiral perturbation theory, though at this preliminary stage our errors for $I=0$ are large. An important outcome of this work is that we are successful in extracting two-pion excited states, which are useful for studying $K\to\pi\pi$ decay, on physical-mass ensembles using GEVP.

Holographic duals of evaporating black holes

We describe the dynamical evaporation of a black hole as the classical evolution in time of a black hole in an Anti-de Sitter braneworld. A bulk black hole whose horizon intersects the brane yields the classical bulk dual of a black hole coupled to quantum conformal fields. The evaporation of this black hole happens when the bulk horizon slides off the brane, making the horizon on the brane shrink. We use a large-D effective theory of the bulk Einstein equations to solve the time evolution of these systems. With this method, we study the dual evaporation of a variety of black holes interacting with colder radiation baths. We also obtain the dual of the collapse of holographic radiation to form a black hole on the brane. Finally, we discuss the evolution of the Page curve of the radiation in our evaporation setups, with entanglement islands appearing and then shrinking during the decreasing part of the curve.

Improved Oscillation Theorems for Even-Order Quasi-Linear Neutral Differential Equations

In this study, our goal was to establish improved inequalities that enhance the asymptotic and oscillatory behaviors of solutions to even-order neutral differential equations. In the oscillation theory of neutral differential equations, the connection between the solution and its corresponding function plays a critical role. We refined these relationships by leveraging the modified monotonic properties of positive solutions and introduced new conditions that ensure the absence of positive solutions, confirming the oscillation of all solutions to the studied equation. Based on the concept of symmetry between the positive and negative solutions of the studied equation, we obtained criteria that guarantee the oscillation of all solutions by excluding positive solutions only. In order to demonstrate the significance of our findings, we examined certain instances of the studied equation and compared them with previous results in the literature.